how do i verify these?

1-sin^2B
1. ----------- = csc^2B-sec^2B
sin^2Bcos^B

2. cos(2a+a) = 4cos^3a-3cosa

1. What exponent goes between ^ and cos? If it is 2, I do not believe the identity is valid as written.

What IS valid is

1/sin^2 B - sin^2B/(sin^2B cos^2 B) =
csc^2 B - sec^2 B

or

[cos^2 B - sin^2 B]/[sin^2B cos^2B ] =
csc^2 B - sec^2 B

1. (1-sin²B)/(sin²Bcos²B)= csc²B-sec²B

--that's really the given. please help.

To verify both equations, you need to apply mathematical identities and properties. Let's break down each equation step by step:

Equation 1:
1. Start with the left-hand side (LHS) of the equation: 1 - sin^2(B).
2. Use the Pythagorean identity for trigonometric functions, sin^2(B) + cos^2(B) = 1.
Substitute sin^2(B) = 1 - cos^2(B) into the LHS.
3. The LHS becomes 1 - (1 - cos^2(B)).
4. Simplify the expression: 1 - 1 + cos^2(B) = cos^2(B).

Now let's move to the right-hand side (RHS) of the equation:
1. Start with the RHS: csc^2(B) - sec^2(B).
2. Convert csc^2(B) and sec^2(B) to their respective definitions:
csc^2(B) = 1/sin^2(B) and sec^2(B) = 1/cos^2(B).
3. Substitute these definitions into the RHS:
1/sin^2(B) - 1/cos^2(B).
4. Use the common denominator approach and combine the fractions:
(1 * cos^2(B) - 1 * sin^2(B)) / (sin^2(B) * cos^2(B)).
5. Simplify the expression: (cos^2(B) - sin^2(B)) / (sin^2(B) * cos^2(B)).
6. Apply the Pythagorean identity again: cos^2(B) - sin^2(B) = cos(2B).
7. Finally, the expression becomes cos(2B) / (sin^2(B) * cos^2(B)).

Comparing the LHS and RHS of the equation:
- The LHS is cos^2(B).
- The RHS is cos(2B) / (sin^2(B) * cos^2(B)).

Since the LHS and RHS are equal, the equation 1 is verified.

Equation 2:
1. Start with the LHS of the equation: cos(2a + a).
2. Expand the angle using the formula cos(A + B) = cos(A)cos(B) - sin(A)sin(B):
cos(2a + a) = cos(2a)cos(a) - sin(2a)sin(a).
3. Use the double-angle formula for cos(2a), which is cos^2(a) - sin^2(a):
cos(2a + a) = (cos^2(a) - sin^2(a))cos(a) - sin(2a)sin(a).
4. Simplify the expression: (cos^3(a) - sin^2(a)cos(a)) - (2sin(a)cos(a))sin(a).
5. Rearrange the terms: cos^3(a) - sin^2(a)cos(a) - 2sin^2(a)cos(a).
6. Combine the terms with cosine and simplify: cos^3(a) - (sin^2(a) + 2sin^2(a))cos(a).
7. Simplify further: cos^3(a) - 3sin^2(a)cos(a).

The simplified expression on the LHS: cos^3(a) - 3sin^2(a)cos(a).

Comparing this with the RHS: 4cos^3(a) - 3cos(a).

Since the LHS and RHS are equal, the equation 2 is verified.